I know that selling things without going through App Store In-App Purchases etc. is not allowed widely. Even apps like Dropbox historically had problems with it and even they use in-app purchases for Dropbox upgrades today.

If I'd develop an app that people donate charities using an online payment system (like Balanced or Stripe), would Apple still take my app down because I'm not going through in-app puchases?

I don't believe that limitation effects Apple Pay transactions, since it's not going through Apple's in-app purchase program. It also wouldn't be subject to Apple's 30% cut. Apple would get a portion of the transaction, but in the form of a fee charged to your credit processing merchant for the service of generating the secure Apple Pay transaction tokens.